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Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 14: Review / Subdivision Ravi Ramamoorthi http://www.cs.columbia.edu/~cs4162 Slides courtesy of Szymon Rusinkiewicz with material from Denis Zorin, Peter Schroder
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To Do Questions of any kind? Discuss this first (main point of lecture) Comments? Interesting assignment since fairly new topics Suggestions for future iterations of course?
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Subdivision Very hot topic in computer graphics today Brief survey lecture, quickly discuss ideas Detailed study quite sophisticated Advantages Simple (only need subdivision rule) Local (only look at nearby vertices) Arbitrary topology (since only local) No seams (unlike joining spline patches)
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Video: Geri’s Game (Pixar website)
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Subdivision Surfaces Coarse mesh & subdivision rule Smooth surface = limit of sequence of refinements [Zorin & Schröder]
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Key Questions How to refine mesh? Where to place new vertices? Provable properties about limit surface [Zorin & Schröder]
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Loop Subdivision Scheme How refine mesh? Refine each triangle into 4 triangles by splitting each edge and connecting new vertices [Zorin & Schröder]
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Loop Subdivision Scheme Where to place new vertices? Choose locations for new vertices as weighted average of original vertices in local neighborhood [Zorin & Schröder]
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Loop Subdivision Scheme Where to place new vertices? Rules for extraordinary vertices and boundaries: [Zorin & Schröder]
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Loop Subdivision Scheme Choose by analyzing continuity of limit surface Original Loop Warren
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Butterfly Subdivision Interpolating subdivision: larger neighborhood 1/21/21/21/2 1/21/21/21/2 -1 / 16 1/81/81/81/8 1/81/81/81/8
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Modified Butterfly Subdivision Need special weights near extraordinary vertices For n = 3, weights are 5 / 12, -1 / 12, -1 / 12 For n = 4, weights are 3/8, 0, -1/8, 0 For n 5, weights are Weight of extraordinary vertex = 1 - other weights
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A Variety of Subdivision Schemes Triangles vs. Quads Interpolating vs. approximating [Zorin & Schröder]
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More Exotic Methods Kobbelt’s subdivision:
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More Exotic Methods Kobbelt’s subdivision: Number of faces triples per iteration: gives finer control over polygon count
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Subdivision Schemes [Zorin & Schröder]
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Subdivision Schemes [Zorin & Schröder]
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Analyzing Subdivision Schemes Limit surface has provable smoothness properties [Zorin & Schröder]
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Analyzing Subdivision Schemes Start with curves: 4-point interpolating scheme -1 / 16 9 / 16 (old points left where they are)
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4-Point Scheme What is the support? v0v0v0v0 v -2 v -1 v1v1v1v1 v2v2v2v2 Step i: v0v0v0v0 v -2 v -1 v1v1v1v1 v2v2v2v2 Step i+1: So, 5 new points depend on 5 old points
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Subdivision Matrix How are vertices in neighborhood refined? (with vertex renumbering like in last slide)
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Subdivision Matrix How are vertices in neighborhood refined? (with vertex renumbering like in last slide) After n rounds:
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Convergence Criterion Expand in eigenvectors of S: Criterion I : | i | 1
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Convergence Criterion What if all eigenvalues of S are < 1? All points converge to 0 with repeated subdivision Criterion II : 0 = 1
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Translation Invariance For any translation t, want: Criterion III : e 0 = 1, all other | i | < 1
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Smoothness Criterion Plug back in: Dominated by largest i Case 1: | 1 | > | 2 | Group of 5 points gets shorter All points approach multiples of e 1 on a straight line Smooth!
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Smoothness Criterion Case 2: | 1 | = | 2 | Points can be anywhere in space spanned by e 1, e 2 No longer have smoothness guarantee Criterion IV : Smooth iff 0 = 1 > | 1 | > | i |
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Continuity and Smoothness So, what about 4-point scheme? Eigenvalues = 1, 1 / 2, 1 / 4, 1 / 4, 1 / 8 e 0 = 1 Stable Translation invariant Smooth
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2-Point Scheme In contrast, consider 2-point interpolating scheme Support = 3 Subdivision matrix = 1/21/21/21/2 1/21/21/21/2
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Continuity of 2-Point Scheme Eigenvalues = 1, 1 / 2, 1 / 2 e 0 = 1 Stable Translation invariant Smooth X Not smooth; in fact, this is piecewise linear
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For Surfaces… Similar analysis: determine support, construct subdivision matrix, find eigenstuff Caveat 1: separate analysis for each vertex valence Caveat 2: consider more than 1 subdominant eigenvalue Points lie in subspace spanned by e 1 and e 2 If | 1 | | 2 |, neighborhood stretched when subdivided, but remains 2-manifold Reif’s smoothness condition: 0 = 1 > | 1 | | 2 | > | i |
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Fun with Subdivision Methods Behavior of surfaces depends on eigenvalues (recall that symmetric matrices have real eigenvalues) [Zorin] ComplexDegenerateReal
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Summary Advantages: Simple method for describing complex, smooth surfaces Relatively easy to implement Arbitrary topology Local support Guaranteed continuity Multiresolution Difficulties: Intuitive specification Parameterization Intersections [Pixar]
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